A303656 -id:A303656 - cp's OEIS frontend

A308734 Number of ordered ways to write n as (2^a3^b)^2 + (2^c5^d)^2 + x^2 + y^2, where a,b,c,d,x,y are nonnegative integers with x <= y.

0, 1, 1, 1, 2, 3, 3, 1, 3, 5, 2, 3, 4, 4, 5, 1, 4, 8, 4, 4, 8, 8, 4, 3, 8, 7, 7, 6, 5, 13, 6, 1, 10, 11, 7, 7, 10, 9, 9, 5, 7, 18, 7, 5, 14, 11, 6, 3, 10, 11, 9, 8, 7, 15, 9, 4, 14, 12, 5, 10, 9, 10, 11, 1, 11, 19, 10, 6, 17, 21, 6, 8, 14, 12, 13, 7, 14, 21, 7, 4

Offset: 1

Zhi-Wei Sun, Jun 21 2019

nonn

Four-square Conjecture: a(n) > 0 for all n > 1.
This is much stronger than Lagrange's four-square theorem. We have verified a(n) > 0 for all n = 2..10^9.
Note that 16265031 cannot be written as (2^a*3^b)^2 + (2^c*3^d)^2 + x^2 + y^2 with a,b,c,d,x,y nonnegative integers.
a(n) > 0 for 1 < n <= 10^10. - Giovanni Resta, Jun 28 2019
I promise to offer 2500 US dollars as the prize for the first correct proof of the Four-square Conjecture. - Zhi-Wei Sun, Jul 09 2019
Jiao-Min Lin (a student at Nanjing University) has verified a(n) > 0 for all 1 < n <= 1.6*10^11. - Zhi-Wei Sun, Jul 30 2022

			a(2^(2k+1)) = 1 with 2^(2k+1) = (2^k*3^0)^2 + (2^k*5^0)^2 + 0^2 + 0^2.
a(2^(2k+2)) = 1 with 2^(2k+2) = (2^k*3^0)^2 + (2^k*5^0)^2 + (2^k)^2 + (2^k)^2.
a(3) = 1 with 3 = (2^0*3^0)^2 + (2^0*5^0)^2 + 0^2 + 1^2.
a(5) = 2 with 5 = (2^0*3^0)^2 + (2^1*5^0)^2 + 0^2 + 0^2 = (2^1*3^0)^2 + (2^0*5^0)^2 + 0^2 + 0^2.
a(11) = 2 with 11 = (2^0*3^0)^2 + (2^0*5^0)^2 + 0^2 + 3^2 = (2^0*3^1)^2 + (2^0*5^0)^2 + 0^2 + 1^2.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
Soumyarup Banerjee, On a conjecture of Sun about sums of restricted squares, J. Number Theory 256 (2024), 253-289.
Zhi-Wei Sun, Refining Lagrange's four-square theorem, J. Number Theory 175 (2017), 167-190.
Zhi-Wei Sun, Restricted sums of four squares, Int. J. Number Theory 15 (2019), 1863-1893.
Zhi-Wei Sun, Various Refinements of Lagrange's Four-Square Theorem, Westlake Number Theory Symposium (Nanjing University, China, 2020).
Zhi-Wei Sun, New Conjectures in Number Theory and Combinatorics (in Chinese), Harbin Institute of Technology Press, 2021. (See Conjecture 5.16.)

Cf. A000079, A000118, A000290, A000244, A000351, A271518, A281976, A303656, A308566, A308584, A308621, A308623, A308640, A308641, A308644, A308656, A308661, A308662.

SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]];
tab={};Do[r=0;Do[If[SQ[n-4^a*9^b-4^c*25^d-x^2],r=r+1],{a,0,Log[4,n]},{b,0,Ceiling[Log[9,n/4^a]]-1},
{c,0,Log[4,n-4^a*9^b]},{d,0,Log[25,(n-4^a*9^b)/4^c]},{x,0,Sqrt[(n-4^a*9^b-4^c*25^d)/2]}];tab=Append[tab,r],{n,1,80}];Print[tab]

A305232 Number of ordered ways to write 2n+1 as p + binomial(2k,k) + 2binomial(2m,m), where p is an odd prime, and k and m are nonnegative integers.

Original entry on oeis.org

0, 0, 1, 2, 3, 3, 3, 4, 3, 5, 4, 5, 6, 5, 4, 4, 6, 6, 4, 5, 4, 6, 6, 6, 7, 6, 6, 4, 5, 6, 6, 8, 5, 5, 6, 5, 7, 9, 8, 5, 8, 9, 6, 9, 7, 8, 6, 6, 4, 7, 8, 7, 7, 4, 8, 10, 9, 7, 8, 9, 5, 7, 6, 5, 7, 7, 7, 3, 6, 7, 7, 9, 6, 9, 6, 9, 9, 7, 7, 8, 9, 6, 5, 8, 10, 10, 6, 8, 7, 9

Offset: 1

Zhi-Wei Sun, May 27 2018

nonn

The first value of n > 2 with a(n) = 0 is 15212443837. Neither 2*15212443837 + 1 = 30424887675 nor 2*15657981007 + 1 = 31315962015 can be written as the sum of a prime, a central binomial coefficient and twice a central binomial coefficient.

			a(3) = 1 since 2*3 + 1 = 7 = 3 + binomial(2*1,1) + 2*binomial(2*0,0) with 3 an odd prime.
a(368233372) = 1 since 2*368233372 + 1 = 736466745 = 735761311 + binomial(2*11,11) + 2*binomial(2*0,0) with 735761311 an odd prime.
a(5274658504) = 1 since 2*5274658504 + 1 = 10549317009 = 10549316083 + binomial(2*6,6) + 2*binomial(2*0,0) with 10549316083 an odd prime.
a(8722422187) = 1 since 2*8722422187 + 1 = 17444844375 = 17444844367 + binomial(2*2,2) + 2*binomial(2*0,0) with 17444844367 an odd prime.
a(10296844792) = 1 since 2*10296844792 + 1 = 20593689585 = 20593688659 + binomial(2*6,6) + 2*binomial(2*0,0) with 20593688659 an odd prime.

Zhi-Wei Sun, Table of n, a(n) for n = 1..100000
Zhi-Wei Sun, Mixed sums of primes and other terms, in: D. Chudnovsky and G. Chudnovsky (eds.), Additive Number Theory, Springer, New York, 2010, pp. 341-353.
Zhi-Wei Sun, Conjectures on representations involving primes, in: M. Nathanson (ed.), Combinatorial and Additive Number Theory II, Springer Proc. in Math. & Stat., Vol. 220, Springer, Cham, 2017, pp. 279-310. (See also arXiv:1211.1588 [math.NT], 2012-2017.)

Cf. A000040, A000984, A303540, A303656, A303702, A303821, A303934, A304034, A304081, A305030.

tab={};Do[r=0;k=0;Label[aa];k=k+1;If[Binomial[2k,k]>=2n+1`,Goto[cc]];m=0;Label[bb];If[2*Binomial[2m,m]>=2n+1-Binomial[2k,k],Goto[aa]]; If[PrimeQ[2n+1-Binomial[2k,k]-2*Binomial[2m,m]],r=r+1];m=m+1;Goto[bb];Label[cc];tab=Append[tab,r],{n,1,90}];Print[tab]

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A308734 Number of ordered ways to write n as (2^a3^b)^2 + (2^c5^d)^2 + x^2 + y^2, where a,b,c,d,x,y are nonnegative integers with x <= y.

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A305232 Number of ordered ways to write 2n+1 as p + binomial(2k,k) + 2binomial(2m,m), where p is an odd prime, and k and m are nonnegative integers.

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cp's OEIS Frontend

A308734 Number of ordered ways to write n as (2^a*3^b)^2 + (2^c*5^d)^2 + x^2 + y^2, where a,b,c,d,x,y are nonnegative integers with x <= y.

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A305232 Number of ordered ways to write 2*n+1 as p + binomial(2k,k) + 2*binomial(2m,m), where p is an odd prime, and k and m are nonnegative integers.

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A308734 Number of ordered ways to write n as (2^a3^b)^2 + (2^c5^d)^2 + x^2 + y^2, where a,b,c,d,x,y are nonnegative integers with x <= y.

A305232 Number of ordered ways to write 2n+1 as p + binomial(2k,k) + 2binomial(2m,m), where p is an odd prime, and k and m are nonnegative integers.